Question

2. Table 6, "Users' Stated Marital Preferences," is taken from "Preferences and Choice Constraints in Martial Sorting: Evidence from Korea," by Soohyung Lee. This data is from users of a Korean match-making website. Users are asked to state their firs, second, and third highest priorities in a potential spouse. Table 6 reports the results separately for male and female users. Let X, = 1 if individual i said appearance was the highest priority in a potential spouse, and Xi = 0 otherwise. (a) From the table, what is the sample mean of X among men? among women? (b) Use your answer to part (a) to estimate the variance of X conditional on being male, and to estimate the variance of X conditional on being female c) Use your answer to part (b) and the reported sample sizes to compute the standard errors on the sample mean of X among men and the sample mean of X among women. (Reminder: standard error is an estimate of the variance of the sample mean (d) Suppose you wished to construct a confidence interval on E(X|Male) and a confi dence interval on E(XFemale). Why is an asymptotic approach reasonable in this application? (e) Construct a 90-percent confidence interval on E(XMale) (f) Construct a 90-percent confidence interval on E(XFemale) (g) Consider now X,-1 if the individual said he/she would avoid a prospective spouse with no religion, and X, - 0 otherwise. You want to test whether there is dif- ference between E(X|Male) and E(X]Female). What are the null and alternative hypotheses? (h) What is the test statistic (explain how to build it)? What is its approximate dis tribution under Ho? (i) What is the rejection region of the test at 1%? (j) Compute the test statistic and conclude (k) What is the approximate p-value for this test? Table 6: Users' Stated Marital Preferences The Three Most Important Characteristics for a Prospective Spouse Number ofDistribution across prospective spouse's characteristics (Percentage) Occupation Education observations Appearance Personality Religion Age Others and Income Men st priority priority priority 6,334 6,334 5,991 11.02 16.47 23.21 33.71 25.51 15.31 2.01 5.00 6.96 1.97 2.78 3.95 1.36 6.7 10.82 2.65 8.31 23.21 44.57 34.13 20.35 nd rd Women 1 priority 2nd priority priorit 7,539 7,421 7,156 5.07 8.56 23.30 26.82 24.40 16.62 55.64 44.19 21.03 4.42 11.44 8.06 3.32 0.90 3.83 1.82 2.12 7.47 3.143.44 24.41 rd A Prospective Spouse's Religion that a User Avoids Number of Observations 9,458 11,052 Avoiding religiorn None 50.9 50.7 Protestant Catholic Buddhist No religion Other religions 25.0 24.1 Men Women 21.9 22.5 0.3 0.6 0.0 0.0 Prospective Spouse's Residential Area or Hometown that a User Avoids Seoul 0.12 0.29 Gveonggi 1.49 1.52 Gangwon Chungcheong Jeolla Gyeongsang Jeju and Others None 62.21 63.80 2.03 0.98 1.27 0.73 Men Women 0.26 0.13 0.23 0.16 32.39 32.39

Answer 1

Solution

X_i as defined in the question, is dichotomous taking values 0 or 1. So, it has Bernoulli distribution. …… (1)

Then, X ~ B(n, p), ………………………………………………………………………………………………….(2)

where n = number of individuals in the sample and p is estimated by the sample proportion, say p_cap

Back-up Theory

If X ~ B(n, p). i.e., X has Binomial Distribution with parameters n and p, where n = number of trials

and p = probability of one success, then, probability mass function (pmf) of X is given by

p(x) = P(X = x) = (ⁿC_x)(p^x)(1 - p)^{n – x}, x = 0, 1, 2, ……. , n ……………………………………..………..(3)

[This probability can also be directly obtained using Excel Function: Statistical, BINOMDIST……….(3a)

Mean (average) of X = E(X) = µ = np……………………..………………………….....…………………..(4)

Variance of X = V(X) = σ² = np(1 – p)………………………………………………....………….………..(5)

Standard Deviation of X = SD(X) = σ = √{np(1 – p)} ………………………………...…………………...(6)

If X ~ B(n, p), np ≥ 10 and np(1 - p) ≥ 10, then Binomial probability can be approximated by Standard Normal probabilities by Z = (X – np)/√{np(1 - p)} ~ N(0, 1) …………………………………..(7)

Or, equivalently, if pcap = sample proportion, then, Z = (pcap – p)/√{p(1 - p)/n} ~ N(0, 1) ……….…..(7a)

Given X ~ B(n, p),

100(1 - α) % Confidence Interval for the mean of X is: np_cap ± MoE, ..………………………….……. (8)

where

MoE = Z_α/2[SE(X)] ………………………………….....………………………………………….………..(8a)

With Z_α/2 = upper (α/2)% point of N(0, 1),

Now, to work out the solution,

Part (a)

Men

From the given Table – 6, n = 6334 and p_cap = 0.4557.

So, vide (4), sample mean = 6334 x 0.4557 = 2886.4038 Answer 1

Women

From given Table – 6, n = 7539 and p_cap = 0.0507.

So, vide (4), sample mean = 7539 x 0.0507 = 382.2273 Answer 2

Part (b)

Vide (5),

variance for Men is: 6334 x 0.4557 x 0.5443 = 1571.0696 Answer 3

variance for Women is: 7539 x 0.0507 x 0.9493 = 362.8484 Answer 4

Part (c)

Standard error = sqrt(Variance)

SE for Men = 39.64 Answer 5

SE for Women = 19.05 Answer 6

Part (d)

Vide (7), both np and np(1 - p) are greater than 10 for both men and women. Hence, asymptotic approach is reasonable for all further derivations. Answer 7

Part (e)

Vide (8a), MoE for Men = 1.645 x 39.64 = 65.21

So, 90% confidence interval for Men is: 2886.40 ± 65.21

= [2821.19, 2951.61] Answer 8

Part (e)

Vide (8a), MoE for Women = 1.645 x 19.05 = 31.33

So, 90% confidence interval for Women is: 382.23 ± 31.33

= [350.90, 413.56] Answer 9

DONE